As will be discussed in the following subsections and Appendix , these eigensolutions are alternative representations of characteristic modes obtained through transition [1]} and impedance matrix [2]} methods. In (REF ), the eigenvalues \(t_n\)  describing the relative scaling of incident and scattered fields. This concept is illustrated in Fig. REF b, where an object is illuminated by an eigenexcitation consisting of spectrum of plane waves with vector amplitudes \(F̭_n({r}^{\prime })\)  and the scattered field at all locations on the far-field sphere is a scaled copy of the incident plane wave spectrum, , \(t_nF̭_n({r})\) .
